arXiv subjects
Publications and source records attributed to Prince K. Osei.
We construct a higher lattice gauge theory based on the representation of 2-groups described by a category of crossed modules on a lattice model described by path 2-groupoids. Using these lattice gauge representations, an exactly solvable Hamiltonian for topological phases in 3+1 dimensions is constructed. We show that the ground states of this model are topological observables.
Position deformation of a Heisenberg algebra and Hilbert space representation of both maximal length and minimal momentum uncertainties may lead to loss of Hermiticity of some operators that generate this algebra. Consequently, the Hamiltonian operator constructed from these operators are also not Hermitian. In the present paper, with an appropriate positive-definite Dyson map, we establish the Hermiticity of these operators by means of a quasi-similarity transformation. We then construct Hilbert space representations associated with these quasi-Hermitian operators that generate a quasi-Hermitian Heisenberg algebra. With the help of these representations we establish the path integral formulation of any systems in this quasi-Hermitian algebra. Finally, using the path integral of a free particle as an example, we demonstrate that the Euclidean propagator, action, and kinetic energy of this system are constrained by the standard classical mechanics limits.
The (2+1)-dimensional analog self-dual gravity which is obtained via spacetime dimension reduction of the (3+1)-dimensional Holst action without reducing the internal gauge group is studied. A Chern-Simons formulation for this theory is constructed based on the gauge group $SL(2,\CC)_\RR\rcross \Rsix$ and maps the 3d complex self-dual dynamical variable and connection to 6d real variables which combines into a 12d Cartan connection. The Chern-Simons approach leads to a real analogue for the self-dual action based on a larger symmetry group. The quantization process follows the combinatorial quantization method outlined for Chern-Simons theory. In the combinatorial quantization of the phase space the Poisson structure governing the moduli space of flat connections which emerges is obtained using the classical $r$-matrix for the quantum double $D(SL(2,\CC)_\RR)$ viewed as the double of a double $ D(SL(2,\RR)\dcross AN(2))$. This quantum double gives the structure for quantum symmetries within the quantum theory for the model.
In this work, we provide a rigorous definition of ribbon operators in the Semidual Kitaev lattice model and study their properties. These operators are essential for understanding quasi-particle excitations within topologically ordered systems. We show that the ribbon operators generate quasi-particle excitations at the ends of the ribbon and reveal themselves as irreducible representations of the Bicrossproduct quantum group $M(H)=H^{\text{cop}}\lrbicross H$ or $M(H)^{\text{op}}$ depending on their chirality or local orientation.
Position-deformed Heisenberg algebra with maximal length uncertainty has recently been proven to induce strong quantum gravitational fields at the Planck scale (2022 J. Phys. A: Math. Theor.55 105303). In the present study, we use the position space representation on the one hand and the Fourier transform and its inverse representations on the other to construct propagators of path integrals within this deformed algebra. The propagators and the corresponding actions of a free particle and a simple harmonic oscillator are discussed as examples. Since the effects of quantum gravity are strong in this Euclidean space, we show that the actions which describe the classical trajectories of both systems are bounded by the ordinary ones of classical mechanics. This indicates that quantum gravity bends the paths of particles, allowing them to travel quickly from one point to another. It is numerically observed by the decrease in values of classical actions as one increases the quantum gravitational effects.
Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double $D(H)=H\bowtie H^{*\text{op}} $, as an example of a double cross product quantum group. We propose a new version based instead on $M(H)=H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$ as an example of Majid's bicrossproduct quantum group, related by semidualisation or `quantum Born reciprocity' to $D(H)$. Given a finite-dimensional Hopf algebra $H$, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group $H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.
We use factorisations of the local isometry groups arising in 3d gravity for Lorentzian and Euclidean signatures and any value of the cosmological constant to construct associated bicrossproduct quantum groups via semidualisation. In this way we obtain quantum doubles of the Lorentz and rotation groups in 3d, as well as kappa-Poincare algebras whose associated r-matrices have spacelike, timelike and lightlike deformation parameters. We confirm and elaborate the interpretation of semiduality proposed in [13] as the exchange of the cosmological length scale and the Planck mass in the context of 3d quantum gravity. In particular, semiduality gives a simple understanding of why the quantum double of the Lorentz group and the kappa-Poincare algebra with spacelike deformation parameter are both associated to 3d gravity with vanishing cosmological constant, while the kappa-Poincare algebra with a timelike deformation parameter can only be associated to 3d gravity if one takes the Planck mass to be imaginary.